Remark 10 Such determinants have appeared before: in the enumeration of nonintersecting lattice paths and plane partitions [1]}; in the context of random matrices from the classical groups [2]}; as distributions for symmetrized longest increasing subsequences in the work of Baik–Rains [3]}; and have been studied extensively from a Szegő asymptotics point of view in the works of Basor–Ehrhardt [4]}, [5]}, [6]}, [7]}, [8]}, Forrester–Frankel [9]} and Deift–Its–Krasovsky [10]}. Note however that our determinant indexing differs from e.g. [3]} via negation of both indices (alternatively, via conjugation of the variable \(z\)  of the symbol \(f\) ). Further note the two constants \(Z_{sp}, Z_{o}\)  are the same as \(E F_{IV}\)  and \(E F_{III}\)  respectively in [7]}.

